Up to 10,000 crossed at Peshkhabour on Saturday, bringing the total influx since Thursday to 20,000. The UN says the reasons are not fully clear.
The UN agencies, the Kurdish regional government and NGOs are struggling to cope, correspondents say.
It comes as UN chemical weapons inspectors arrived in Damascus on Sunday on a much-delayed mission.
The team will visit three sites over two weeks, including the northern town of Khan al-Assal which is at the centre of allegations of chemical weapons use.
‘War and looting’
The UN High Commissioner for Refugees (UNHCR) says this is one of the biggest single waves of refugees it has had to deal with since the uprising against the rule of Syrian President Bashar al-Assad began in March 2011.
While the reasons remain unclear, there has been a sharp rise in clashes between Syrian Kurds and anti-government Islamist militants.
The charity Save the Children has launched an emergency response to the mass arrival, distributing basic supplies to those waiting to be registered.
“This is an unprecedented influx of refugees, and the main concern is that so many of them are stuck out in the open at the border or in emergency reception areas with limited, if any, access to basic services,” said Alan Paul, Save the Children’s emergency team leader, who is in the area.
“The refugee response in Iraq is already thinly stretched, and close to half of the refugees are children who have experienced things no child should.”
The BBC’s Jim Muir in neighbouring Lebanon says the latest refugees are mainly families and have come from a broad stretch of territory in northern Syria.
They have been taking advantage of a new pontoon bridge over the Tigris.
“There was war and looting and problems…We did not find a morsel [of food], so with our children, we came here,” Abdulkarim Brendar, who trekked across the bridge with his five children, told AFP news agency.
Another refugee, Ahmed Karim, told the agency he left Syria to save his wife and three-week-old baby from dying of starvation.
“There was a shortage of food in the market, and everything became expensive, from bread to gas canisters, and unemployment was spreading,” he said.
Some 150,000 Syrian refugees are already registered in Iraq, of the nearly two million said to have fled Syria in total since the uprising began.
The UN said the latest refugees had come from Aleppo, Hassakeh, Qamishli and other areas of conflict.
On Friday, UNHCR spokesperson Adrian Edwards told reporters in Geneva: “The factors allowing this sudden movement are not fully clear to us.”
The UN said it was working with the Iraqi Kurdistan government and other agencies to establish a camp at nearby Darashakran.
“This should open in two weeks, and our hope is it will relieve pressure,” Mr Edwards said.
The ethnic make-up of the latest wave has not been detailed.
Kurds make up about 10% of the Syrian population and are largely concentrated in the north-east.
They staged their own anti-Assad protests after the Syria conflict began in 2011 and their areas have been run by Kurdish local councils and militia since government forces withdrew last year.
But the Kurdish militias have recently been fighting jihadists of the anti-Assad al-Nusra Front, leaving dozens dead.
The president of Iraqi Kurdistan, Massoud Barzani, recently threatened to intervene to defend the Kurdish population caught up in Syria’s unrest.
He said if Kurds were “under threat of death and terrorism” then Iraqi Kurdistan would be “prepared to defend them”.
Iraqi Kurdistan comprises three provinces in northern Iraq. It has its own military and police force.